nLab 12-manifold

Idea

A 12-manifold is a manifold of dimension 12. (Generally a topological manifold, but it can be specified to a PL manifold or a smooth manifold.)

Examples

• 12-sphere
• real projective space$\mathbb{R}P^12$
• complex projective space$\mathbb{C}P^6$
• complex projective space$\mathbb{H}P^3$

Properties

Proposition

(Hirzebruch signature theorem) For an orientable smooth 12-manifold $M$ with fundamental class $[M]\in H^12(M,\mathbb{Z})\cong\mathbb{Z}$, its signature is given by Pontrjagin numbers as:

$$\sigma(M) =\frac{1}{945}\langle(62p_3-13p_1p_2+2p_1^3)(M),[M]\rangle \in\mathbb{Z}.$$

Related concepts

manifolds in low dimension:

• 2-manifolds

• 3-manifolds

• 4-manifolds

• 5-manifolds

• 6-manifolds

• 7-manifolds

• 8-manifolds

• 9-manifolds

• 10-manifolds

• 11-manifolds

• 12-manifolds

• 13-manifolds

• 14-manifolds

• 15-manifolds

• 16-manifolds

Application of 12-manifolds:

• D=12 supergravity